Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source

In this paper, we consider the quasilinear chemotaxis system {u(t) = del . (D(u)del u) - del . (S(u)del v) + f(u), x is an element of Omega, t > 0, (*) v(t) = del v - v + u, x is an element of Omega, t > 0 (*) in a bounded domain Omega subset of R-n (n >= 2) under zero -flux boundary conditions, where the nonlinearities D, S is an element of C-2 ([0, infinity)) are supposed to generalize the prototypes D(u) = C-D (u + 1)(m-1) and S(u) = C(S)u(u +1)(q-1) with C-D, C-S > 0 and m,q is an element of R, and f is an element of C-1 ([0, infinity)) satisfies f(u) <= r bu(gamma) with r >= 0, b > 0 and gamma > 1. It is shown that if q <{max {m + 2/n-1, m + gamma/2-n - 1/n} for 1 < gamma <= n + 2/n, max {m + 2 gamma/n + 2-1, m/2 + gamma(n + 4)/2(n + 2)-1} for n + 2/n < gamma < n + 2/2, max {m + 2/n - n + 2/n gamma, m/2 + gamma(n + 4)/2(n + 2)-1} for n + 2/2 <= gamma < n + 2, max {m + 1/n, m + gamma/2} for gamma >= n + 2, then (*) unique globally bounded classical solution. (C) 2017 Elsevier Ltd. All rights reserved.